Download Economics 326: Long and Short Run Equilibria and Welfare

Economics 326: Long and Short
Run Equilibria and Welfare
Ethan Kaplan
November 28, 2012
Outline
1. Short and Long Run Equilibria
2. Consumer and Producer Surplus
3. Government Price Setting
4. Government Quantity Setting: Quotas
5. Maximizing Surplus
1
Short and Long Run Equilibria
What is the di¤erence?
1. In the short run, some factors may be …xed for
the …rm - thus the individual …rm supply function
may look di¤erent
2. In the short run, the number of …rms who have
entered may earn pro…ts. In the long run, this will
lead to entry (given the free entry assumption)
which will lead pro…ts to go to zero.
For the most part, we will not focus on (1.). With
(2.), the main di¤erence is that
1. In the short run, the number of …rms (J ) ; is a
…xed exogenous parameter
2. In the long run, the number of …rms (J ), is an
endogenous variable
Lets consider a speci…c problem. Lets try to solve
for a price and quantity determined in a competitive
market. In theory, we could start from a production
function, factor prices, a utility function, income and
prices of goods other than the one in consideration.
We also start with a set of consumers: N and …rms:
J (in the long run J will be an endogenous variable,
in the short run, it will be an exogenous parameter).
– We could then derive Marshallian Demand:
XD (pX ; pY ; I )
– We could then derive Supply Functions:
XS (pX ; w; r)
1. First we would derive input demand functions:
K (p; w; r) ; L (p; w; r)
2. Then we could form supply:
Y (K (p; w; r) ; L (p; w; r))
when pro…ts are positive or price is above average cost: p > AC .
Alternatively, we could:
1. Minimize costs subject to an output constraint
to …nd minimized cost functions:
C (q; w; r)
2. Derive supply as the marginal cost curve above
average cost: @C
@q for p > AC
– We then construct industry supply and market
demand functions
– We then impose market clearing and set supply
equal to demand so that we can solve for price:
N XD;M (pX ; pY ; I ) = JXS (pX ; w; r)
– We then can plug price into either supply or demand to get quantity:
XD (peq
x ; py ; I )
– For short run analysis, we are done. For long
run analysis, we need to …gure out the number
of …rms that enter. We do this by solving for
a number of …rms: J such that pro…ts are zero:
p = AC :
2q 2 (J ) + 20
p (J ) =
q (J )
To make things easy, we will start with individual …rm
cost functions and market demand functions (if we
started with individual demand functions, we would
have to aggregate to market demand):
XD = 100
2pX
C (q ) = 2q 2 + 32
also we assume that the number of …rms in the short
run is J = 8:
First we derive the short run supply curve. This is
the same as the marginal cost curve where price is
above average cost:
@C
= 4q = pX
@q
p
q = X
4
(1)
Now we generate industry supply:
J
X
pX
JpX
8pX
q=
=
=
= 2pX
4
4
j=1 4
Finally, we equate industry supply with market demand and solve …rst for the price:
qS
=
=)
=)
2pX = 100
4px = 100
pX = 25
2pX = qD
and then for quantity by plugging the equilibrium
price, pX ; into either supply or demand:
qS = 2pX = 2 25 = 50
note : qD = 100 2pX = 100
2
25 = 50
why do we get the same answer for qS and qD ?
We should also check that pro…ts are greater than
zero (or price is above average cost):
C (q ) > 0
C (q )
or p >
q
To check this, we …rst have to solve for individual
…rm supply. Since market supply is 50 and there are
1 : Since price
8 …rms, individual supply is 50
=
6
8
4
must be greater than average cost in order to cover
…xed cost and since the …rm will equate price with
marginal cost, in this case we get from equation (1) :
pq
pX = 4q: So 4q must be greater than average cost:
4q
>
=)
=)
=)
=)
2q 2 + 32
q
32
4q > 2q +
q
32
2q >
q
q 2 > 16
q>4
so 4 is the minimum pro…table scale of production for
an optimizing …rm. When individual …rms optimally
choose to produce 4 units of production, they will
get a price to just cover their costs.
In the long run, then, …rms enter until the price drops
so that each …rm is producing 4 units. So the price
drops so that quantity per …rm is 4: We can solve
for this price by using the our expression for optimal
quantity given price (supply) - i.e. we can …gure
out the price that would make individual …rm supply
equal to 4 :
p
qS = X = 4 =) pX = 16
4
Finally, we can use the supply equation to determine
how many …rms will enter in order for the price to
reach 16 :
qS
4J
2
2.1
=
=
=)
100 2pX
100 32 = 68
J = 17
Producer and Consumer Surplus
Producer Surplus
Producer Surplus is easier to de…ne:
(p; y0) = py0
c (y0 ) :
Can give two graphical interpretations:
1. Rewrite as
"
(p; y0) = y0 p
#
c (y0 )
:
y0
Pro…t equals rectangle of quantity times (p - Av.
Cost)
2. Remember:
f (x) = f (0) +
Rewrite pro…t as
p
=
Z y
0
0
0+p
p
Z y
0
0
Z x
1dy
c0y (y ) dy
0
fx0 (s) ds:
c (0) +
Z y
0
0
c0y (y ) dy =
c (0) :
Producer surplus is area between price and marginal cost (minus …xed cost)
3
Welfare: Consumer Surplus
Evaluate welfare e¤ects of price change from p0 to
p1
Proposed measure:
E (p0; u)
E (p1; u)
Can rewrite expression above as
E (p0; u)
E (p1; u) =
E (0; u) +
E (0; u) +
=
What is
@E(p;u)
@p ?
Z p
0 @E (p; u)
@p
0
Z p
1 @E (p; u)
Z p
0 @E (p; u)
p1
@p
0
dp
@p
!
dp
!
dp
Remember the envelope theorem:
h
i
@ pX XH + pY YH
@E
=
= XH
@pX
@pX
Result:
@e(p; u)
= XH (p; u)
@p
Welfare mesure is integral of area to the side of Hicksian compensated demand
– Note that this is a problem to estimate empirically because we don’t observe Hicksian Demand.
Show graph.
4
Government Price Setting
What happens when the government sets a price
‡oor (minimum wage)?
– Not Binding - Nothing
– Binding - quantity transacted in the market is
equal to the minimum of supply and demand at
the price ‡oor
What happens when the government sets a price ceiling (rent cap)?
– Not Binding - Nothing
– Binding - quantity transacted in the market is
equal to the minimum of supply and demand at
the price ceiling
Show graphs
5
Government Quantity Setting
What happens when the government sets a quantity
restriction (i.e. on imports or on production)
– Not Binding - Nothing
– Binding - Quantity is produced at that level and
price is Demand-determined
6
Puzzle
Minimum Wage:
– Empirical estimates of the impact of a rise in the
minimum wage on employment are small.
– Since with a binding minimum wage, there should
be an excess supply of labor, employment should
be demand-determined.
– Thus the small response of employment to the
wage tells us that labor demand is relatively inelastic for low-wage workers.
Immigration:
– Empirical estimates of an increase in immigration
is like a shifting out of the labor supply curve.
– The reaction of the wage should depend upon
the elasticitity of labor demand.
– Empirical estimates suggest that immigration has
a small impact even on wages of non-high school
graduates.
– This suggests that labor demand elasticities are
high!
– Contradiction!!
What’s the resolution of this paradox?
– Economists aren’t sure. A couple of possibilities
lie in two areas of economics
1. Search theory (where …rms and workers - or
…rms and consumers) have to …nd eachother
by searching - in these models, supply doesn’t
equal demand.
2. Behavioral theory (where wages are determined
by beliefs and these beliefs shift when prices
change)
7
Maximizing Surplus
The sum of consumer plus producer surplus is maximized at the market clearing price.
We can see this graphically.
Consumer and producer surplus are used standardly
in policy analysis but:
1. Weighs rich people more than poor people (greater
willingness to pay)
2. Weighs producers more than most consumers
3. Uses the area under the Marshallian instead of
the Hicksian as a measure of Consumer Surplus
Question: what happens with rent control? Show
graph.